Several notes on "order of operations" have appeared - and I'm getting abit weary of this non-subject. So I have resent my reply below to similarmessages on other mail lists.-------------------------------------------------------------Whenever I see messages about "order of operations" my first reactionis that this topic doesn't belong on a school mathematics mail list. If you are spending time teaching about order of operations you arewasting the students time on a non-mathematical topic - no useful mathideas are being developed or reinforced. Order of Operations has onlyto do with various versions of conventional laziness when writing mathexpressions. Before high school, and maybe not even then, there areno situations where expressions get so long and complicated that itwould be unwieldy to write them in correct standard form with allparentheses and multiplications appearing rather than being impliedaccording to some calculator convention. Just because the calculatorallows us to be lazy in writing expressions, that doesn't mean it's a goodidea to do it, and especially not in elementary and middle schools.

By dropping parentheses and multiplication signs much is lost forbeginners. The importance of multiplication is diminished by lettingit disappear from expressions - a strange way to honor itssignificance! It also tends to hide from the user of a calculator thefact that a calculator cannot add or multiply more than 2 numbers atone time. When an expression in calculator shorthand is input into acalculator the calculator internally must essentially put in all thoseommitted parentheses in order to parse the expression and so decideprecisely in what order it will have to proceed to evaluate theexpression. Dropping parentheses deemphasizes (until it disappears)the sense of the sequence of time-steps involved in the calculation. After all, the correct definition of a math expression is that it is a"description of how to calculate something" -- so it will take anorderly collection of steps in time to actually carry out theexpression evaluation. An expression is not, as sometimes said, anumber or a funny name for a number.

Another bad effect of dropping multiplication signs is that, if you do it,then inorder to reduce the likelihood of confusion, all variables or names ofmath objects tend to be restricted to single letters (because AB is assumedto be a product, not a single name). This is a badidea, especially for beginners. Names of variables or quantitiesshould whenever possible be recognizable mnemonic shorthands so thatsomeone can open the book to a page and immediately guess what theexpressions stand for, e.g. ageDad, ageSon, Wt, HtA, HtB, LengRope,AreaRec, distPQ, vol, AveX, NumCows, PxTicket. (Math object namesshould not be exactly the same as the ordinary words used in thosesituations - that is confusing too.)

My pedagogical rule: Never introduce new shorthand notations untilthings begin to get complicated enough so that the shorthand might infact be useful, and only after the student has thoroughly understoodthe ideas involved and has practiced enough to be fluent in thecurrent notation usage.

Calculator shorthand (conventional order of operations) is exactly that, asometimes convenient abbreviation of the correct and unambiguous standardmethod of writing math expressions. In education its use is more harmfulthan helpful.-- Ladnor GeissingerMath Dept, CB 3250 Phillips HallUniv of North CarolinaChapel Hill NC 27599